Why are there dark patches in my rendered drop diamonds?
It would be easy if you uploaded the blend file, or drawed where are the dark patches, because I can't see them in your image, but I simply think that the "Dark Patches" you mentioned are simply reflections from the area around your scene that has not a background. Think about it, the left and right portion of your scene doesn't have anything for the ring to reflect, if you are not using a HDRI, which I think you're not, then the dark patches are simply reflections of empty areas.

As for the shader you have, I don't know why you have 4 different colored glass shaders adding on top of each other, might be something to do with that, it should be easier to reacreate the same material with a Principled BSDF shader.

Also your top image is a rendered one, in which engine you are rendering? Cycles or Internal? That might change things as well, your rendering options like bounces can affect that kind of thing too, that's why it's much easier if you uploaded your blend file.

But I hope this answer helped you,